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Question: If a, b, c, d are distinct integers in A.P. such that d = a<sup>2</sup> + b<sup>2</sup> + c<sup>2</s...

If a, b, c, d are distinct integers in A.P. such that d = a2 + b2 + c2 then a + b + c + d is –

A

0

B

1

C

2

D

None of these

Answer

2

Explanation

Solution

Let t be a common difference of A.P. then

b = a + t

c = a + 2t

d = a + 3t

̃ d = a2 + b2 + c2

̃ a + 3t = a2 + (a + t)2 + (a + 2t)2

̃ 5t2 + 3 (2a – 1) t + 3a2 – a = 0 …..(1)

Q t is real ̃ D ³ 0

̃ 9 (2a – 1)2 – 4(5) (3a2 – a) ³ 0

̃ 24 a2 + 16a – 9 £ 0

̃ −13\frac{- 1}{3} – 7012\frac{\sqrt{70}}{12} < a < −13\frac{- 1}{3} + 7012\frac{\sqrt{70}}{12}

̃ a = –1, 0 [Q a is integer]

When a = 0 form (1) we get t = 0, 3/5 reject both values Q t must be non zero integer.

When a = –1 form (1) t = 1, 45\frac{4}{5}

̃ t = 1 a + b + c + d = –1 + 0 + 1 + 2 = 2.